Led by technological advances, the field of MRI has evolved considerably over the last two decades. In recent years, compressive sensing (CS) has attracted considerable attention in the scientific community and demonstrated notable impact on several applications, including MRI. CS exploits the underlying sparsity of the image and allows image recovery from measurements that are far fewer than the ambient degrees of freedom. Recent studies have shown that the combination of CS and parallel MRI (pMRI) is effective in improving image quality and reducing acquisition time by achieving high acceleration rates for both static and dynamic applications.
An effective application of CS typically has three major requirements: the image is sparse in some transform domain, the undersampling artifact is incoherent (noise like) in the sparsifying transform domain, and the image is recovered by a nonlinear method that enforces both image sparsity and data consistency. For MRI, the first requirement is generally met as most MRI images are compressible in an appropriate transform domain, e.g., discrete Wavelet domain. In recent years, significant strides have been made towards the third requirement by developing fast recovery algorithms for large imaging problems. For the second requirement, the high degree of incoherence is generally achieved by employing either Cartesian patterns with random or pseudo-random sampling or non-Cartesian patterns. Although non-Cartesian sampling methods allow far greater flexibility in designing low-coherence sampling patterns, such sampling schemes are highly sensitive to system imperfections and have found limited use in clinical practice.
Even though random sampling generally provides a high degree of incoherence, such sampling patterns can generate inconsistent results due to excessively large gaps or clustering in the sampling pattern. The large gaps lead to high g-factor for pMRI, i.e., ill conditioning of the underlying inverse problem, and the clustering leads to high correlation among k-space data samples, leading to reduced acquisition efficiency. In contrast, pseudo-random sampling provides a high degree of incoherence while regulating the gaps between samples to a nearly uniform size. For MRI applications, empirical evidence indicates that pseudo-random sampling methods tend to generate superior results compared to random sampling methods. For purely spatial MRI applications, Poisson-disk sampling (PDS) remains a popular pseudo-random sampling scheme. Also, PDS can easily incorporate variable sampling density to preferentially sample center regions of k-space with high SNR. PDS, however, cannot be readily extended to heterogeneous domains because it does not allow incorporating domain-specific constraints. For example, in k-t domain, PDS does not provide a mechanism to maintain a constant temporal resolution by fixing the number of samples (readouts) in each time frame. Therefore, in order to effectively apply CS to MRI in heterogeneous domains, often random sampling is employed, which has the tendency to generate inconsistent results.